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I am wondering if, at all, the wave equation, or the heat equation, is ever used in engineering? I think this equation is of theoretical, rather than, practical interest. For example, by solving the wave equation over a circular membrane we get all the possible standing waves on it. This is a theoretical result. But when engineers construct things do they ever rely on this equation?

I would like to see an explicit example in engineering when the heat equation (or wave equation) PDE is set up and explicitly solved by the eigenvalue expansion.

Air
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Nicolas Bourbaki
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    It looks like you're asking if engineers ever use PDEs. It should be trivially obvious that the answer is, yes, sometimes engineers use PDEs. Do you have some specific interest in the wave equation? Why do you mention the heat equation? Is there a reason you specifically ask about when engineers "construct things" as opposed to design or testing? What do you mean by "rely on" this equation (presumably the wave equation)? There are far too many ambiguities in this question. Please [edit] to clarify. – Air Jun 29 '15 at 16:26
  • "It should be trivially obvious that the answer is, yes, sometimes engineers use PDEs.":

    Why should it be obvious when most actual engineers do not use any advanced mathematics?

     – Nicolas Bourbaki Jun 30 '15 at 04:40
  • @NicolasBourbaki, What is engineering problem you need help? To me the above is a statement. Are you looking for help with PDE, heat equations or eigenvalue expansion. – Mahendra Gunawardena Jun 30 '15 at 12:17
  • Please don't rewrite your question in a way that invalidates existing answers. – Air Jun 30 '15 at 15:27
  • @MahendraGunawardena I am only looking for an explict example in actual engineering, when legit engineers who are building something, have to solve by eigenfunction expansion. Do you know of some actual real-life project had relied on an eigenfunction expansion in order to properly build the project? – Nicolas Bourbaki Jul 01 '15 at 05:14
  • @NicolasBourbaki, In my long engineering career I have never used PDE explicitly, but use tools that use PDE as underlining building blocks. As an engineer I spend time trying to solve the engineering problem. I hardly worry about the underling PDE is equations. I leave it to the aspiring mathematicians and physicist figure out. – Mahendra Gunawardena Jul 01 '15 at 11:11
  • @MahendraGunawardena Thank you for your response. That is what I was curious to see. Namely, if actual engineers use eigenfunction expansions, other methods, ect. in their projects. – Nicolas Bourbaki Jul 02 '15 at 03:32
  • The simple answer is as an Engineer I have never had to used eigenvalue expansion explicitly. But I do use software tools where eigenvalue expansion is use to solve an higher level engineering problem. – Mahendra Gunawardena Jul 02 '15 at 11:14
  • I think that you are going to have to provide your definition of "Engineer". Engineers (licensed) cover a wide range of experience and ability. Even more people are Engineers (unlicensed) because it either isn't required or doesn't apply to their field. And then there are the people who may or may not fit into those categories even if their title says "Engineer". It seems very likely that someone in one of those groups has used PDEs in a non-theoretical way. – hazzey Jul 02 '15 at 20:32
  • @hazzey I am not sure how to properly define "engineer". I just mean the naive version. Somebody who stereotypically is the person who builds, or constructs something, in real life projects. Or perhaps the project coordinator. Let us take you for example. You say you are a "structural engineer". Do you ever, in your line or work, solve PDE by eigenfunction expansion? – Nicolas Bourbaki Jul 03 '15 at 07:23
  • I appreciate that you clarified your question. I haven't voted to reopen because I don't think the clarified question is productive or reasonably scoped; however, any user with the necessary privilege can vote to reopen. The best place for you to express your concerns about this question or my actions would be on Meta Engineering. If you remain unsatisfied, there is a "contact us" link in the site footer where you can escalate the issue to staff. I recommend trying Meta first. – Air Jul 04 '15 at 04:20

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The basic partial differential equations are relied on in many ways.

  1. If I have something already in my book of equations (Roark's Formula's), then I use that book for solving of the plate. These results were derived from the original equation and put in a more useful format

  2. If I don't have it in my book (for example, recently I had to find the equation for a Navier-style plate with two free edges and two simply supported edges with a concentrated rectangular load in the center), then I turn back to the original partial differential equation - the plate equation. In this case, I found a solution online using scholar article searches.

  3. No matter the solution that arrives, certain aspects are maintained as best as possible - conservation of energy, momentum, mass, etc. Each equation usually has a specific aspect that needs to be observed and utilized in even a numerical or FEA solution.

Mark
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